Natalia Janson, Loughborough University

<H3>Control of noise-induced oscillations</H3>

We address the problem of control of oscillations in nonlinear
systems that are induced merely by external noise. Such oscillations can 
have features
that are typical for periodic self-sustained oscillations influenced by 
noise. The
control method we propose is to apply feedback to the system that is 
proportional to
the difference between the current state of the system and its state some 
$\tau$ time
units before. It is applied to two qualitatively distinct types of systems: 
first, to
a self-oscillator below Andronov-Hopf bifurcation, that is to a system with 
a potential
to oscillate autonomously which is not realized under the given control 
parameters; and,
second, to an excitable system that, being pushed with a small but 
sufficient strength,
produces one pulse of an essentially constant shape. The second system is 
often used as
a simplified model of a neuron. In both cases we demonstrate that by 
variation of time
delay $\tau$ one can effectively change regularity and the most probable 
(and also
average) period of stochastic oscillations. The entrainment of the most 
probable period
of oscillations by time delay is discovered. We give explanations of the 
phenomena
observed and provide a theory for the system near bifurcation.